Best Known (99−47, 99, s)-Nets in Base 27
(99−47, 99, 204)-Net over F27 — Constructive and digital
Digital (52, 99, 204)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 34, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 11 and N(F) ≥ 96, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- digital (18, 65, 108)-net over F27, using
- net from sequence [i] based on digital (18, 107)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 18 and N(F) ≥ 108, using
- F3 from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F27 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 18 and N(F) ≥ 108, using
- net from sequence [i] based on digital (18, 107)-sequence over F27, using
- digital (11, 34, 96)-net over F27, using
(99−47, 99, 370)-Net in Base 27 — Constructive
(52, 99, 370)-net in base 27, using
- t-expansion [i] based on (43, 99, 370)-net in base 27, using
- 9 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
- 9 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(99−47, 99, 858)-Net over F27 — Digital
Digital (52, 99, 858)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2799, 858, F27, 47) (dual of [858, 759, 48]-code), using
- 117 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 34 times 0, 1, 49 times 0) [i] based on linear OA(2790, 732, F27, 47) (dual of [732, 642, 48]-code), using
- construction XX applied to C1 = C([727,44]), C2 = C([0,45]), C3 = C1 + C2 = C([0,44]), and C∩ = C1 ∩ C2 = C([727,45]) [i] based on
- linear OA(2788, 728, F27, 46) (dual of [728, 640, 47]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,44}, and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(2788, 728, F27, 46) (dual of [728, 640, 47]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,45], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(2790, 728, F27, 47) (dual of [728, 638, 48]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,45}, and designed minimum distance d ≥ |I|+1 = 48 [i]
- linear OA(2786, 728, F27, 45) (dual of [728, 642, 46]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,44], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,44]), C2 = C([0,45]), C3 = C1 + C2 = C([0,44]), and C∩ = C1 ∩ C2 = C([727,45]) [i] based on
- 117 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 34 times 0, 1, 49 times 0) [i] based on linear OA(2790, 732, F27, 47) (dual of [732, 642, 48]-code), using
(99−47, 99, 455329)-Net in Base 27 — Upper bound on s
There is no (52, 99, 455330)-net in base 27, because
- 1 times m-reduction [i] would yield (52, 98, 455330)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 187 783482 881521 699857 484431 321192 729864 764257 883934 267890 148542 347960 809083 044404 563549 203105 734809 800809 202084 057847 417064 563457 385202 328441 > 2798 [i]